Fix any \(\bar{\mathsf {P}}\)  with \(\mathcal {E}\mathrm {nt}(\bar{\mathsf {P}}|\Pi _n)<\infty \) . We proceed by approximating the unbounded kernel \(\bar{\kappa }_n\)  with bounded ones. For every \(\varepsilon >0\) , we introduce the regularized jump kernel \(\bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta )\)  over \(\Gamma \)  defined by
\(\bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta ):=\frac{1}{1+\varepsilon \nu (\mathcal {T})\eta (\mathcal {T})} \bar{\kappa }_{n}(\nu ,\mathrm {d}\eta ).\) 
In terms of birth/death kernels this can be rewritten as
\(\bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta ) = n \int _{\mathcal {T}}\delta _{\nu + \genfrac{}{}{}2{1}{n} \delta _x}(\mathrm {d}\eta )\, \kappa ^{+,\varepsilon }_{\nu }(\mathrm {d}x) + n \int _{\mathcal {T}} \delta _{\nu -\genfrac{}{}{}2{1}{n} \delta _x}(\mathrm {d}\eta )\,\kappa ^{-,\varepsilon }_{\nu }(\mathrm {d}x),\) 
where
\(\kappa ^{\pm ,\varepsilon }_{\nu }:=\frac{1}{1+\varepsilon \nu (\mathcal {T})(\nu (\mathcal {T})\pm \genfrac{}{}{}2{1}{n})}\kappa ^{\pm }_{\nu }.\) 
Note that
\(\sup _{\nu \in \Gamma } \kappa ^{\pm ,\varepsilon }_{\nu } < \infty \qquad \mbox{ for all }\varepsilon >0.\) 
Correspondingly, we denote \(\vartheta _{\mathsf {P}}^{\pm ,\varepsilon }\) , \(\Theta ^{n,\pm ,\varepsilon }_{\mathsf {P}}\) , \(Q_{n,\varepsilon }\) , \(Q_{n,\varepsilon }^*\) , \(\mathcal {R}_{n,\varepsilon }\) , \(\mathcal {D}_{n,\varepsilon }\) , \(\mathcal {I}_{n,\varepsilon }\) , \((\mathsf {FKE}_{n,\varepsilon })\)  as the relevant quantities, operators, functionals and forward Kolmogorov equations induced by \(\kappa ^{\pm ,\varepsilon }_{\nu }\) . We will first show existence of gradient-flow solutions for the regularized problems, i.e. curves such that \(\mathcal {I}_{n,\varepsilon }=0\) , and then construct an appropriate limit curve as \(\varepsilon \rightarrow 0\) .
Thus, fix any \(\varepsilon >0\) . Due to the bound (REF ) it is clear that \(Q_{n,\varepsilon }\)  is a bounded operator since
\(\sup _{\nu \in \Gamma } \int _{\Gamma } \bar{\kappa }_n^{\varepsilon }(\nu ,\mathrm {d}\eta ) < \infty .\) 
Moreover, since the prefactor \(\nu (\mathcal {T})\eta (\mathcal {T})\)  is symmetric under swapping of \(\nu \)  and \(\eta \) , it straightforward to verify that \(\bar{\kappa }_{n}^{\varepsilon }\)  is still reversible with respect to the same invariant measure \(\Pi _n\) , i.e. we have
\( \Pi _n(\mathrm {d}\nu )\bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta ) = \Pi _n(\mathrm {d}\eta )\bar{\kappa }_{n}^{\varepsilon }(\eta ,\mathrm {d}\nu ). \) 
The triple \((\Gamma ,\Pi _n,\bar{\kappa }_{n}^{\varepsilon })\)  therefore satisfies the assumptions of [1]}. Keeping in mind the difference in definitions of \(\Psi ^*\)  due to extra the factor 2, by [1]} there exist a unique curve \(U^{\varepsilon }\in C^1([0,T],L^1(\Gamma ,\Pi _n))\)  such that \(U_0=\mathrm {d}\bar{\mathsf {P}}/\mathrm {d}\Pi _n\) , and
\(\left\lbrace \begin{aligned}\partial _t U_t(\nu ) &= \int _{\Gamma } (U_t(\eta )-U_t(\nu ))\bar{\kappa }_n^{\varepsilon }(\nu ,\mathrm {d}\eta ), \qquad \mbox{for a.e.\ $t\in [0,T]$}, \\\mathcal {E}\mathrm {nt}(\mathsf {P}_0|\Pi _n)- \mathcal {E}\mathrm {nt}(\mathsf {P}_T|\Pi _n)&= \int _0^T \int _{\Gamma \times \Gamma } \Psi \left(U_t(\eta )-U_t(\nu ) \right) \sqrt{U_t(\nu )U_t(\eta )} \, \Pi _n(\mathrm {d}\nu ) \, \bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta ) \, \mathrm {d}t\\& \quad + \int _{\Gamma \times \Gamma } \left(\sqrt{U_t(\eta )}-\sqrt{U_t(\nu )}\right)^2 \Pi _n(\mathrm {d}\nu ) \, \bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta ) \, \mathrm {d}t,\end{aligned}\right.\) 
with \(\mathsf {P}_t:=U_t \Pi _n\)  as usual. In particular the entropy \(\mathcal {E}\mathrm {nt}(\mathsf {P}|\Pi _n)\)  decreases along the solution and hence
\(\sup _{t\in [0,T]} \mathcal {E}\mathrm {nt}(\mathsf {P}_t|\Pi _n) \le \mathcal {E}\mathrm {nt}(\bar{\mathsf {P}}|\Pi _n).\) 
By evenness of \(\Psi \) , symmetry of \(\Pi _n \bar{\kappa }_n^{\varepsilon }\)  and the identity (REF ), we can express for any \(U\)  after substituting for \(\bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta )\)
\(\frac{1}{2} \int _{\Gamma \times \Gamma } &\Psi \left(U(\eta )-U(\nu ) \right) \sqrt{U(\nu )U(\eta )} \, \Pi _n(\mathrm {d}\nu ) \, \bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta ) \\&= \int _{U(\eta )>0,U(\nu )>0}\phi \left(\sqrt{U(\nu )/U(\eta )}\right)\sqrt{U(\eta )U(\nu )} \, \Pi _n(\mathrm {d}\nu ) \, \bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta )\\&= \int _{U(\nu +n^{-1}\delta _x)>0,U(\nu )>0} \phi \left(\frac{\mathrm {d}\vartheta _{\mathsf {P}}^{+,\varepsilon }}{\mathrm {d}\Theta ^{+,n,\varepsilon }_{\mathsf {P}}} \right)\sqrt{U(\nu +\genfrac{}{}{}2{1}{n}\delta _x)U(\nu )} \, \Pi _n(\mathrm {d}\nu ) \, \kappa ^{+,\varepsilon }_{\nu }(\mathrm {d}x) \\&\qquad + \int _{U(\nu -n^{-1}\delta _x)>0,U(\nu )>0} \phi \left(\frac{\mathrm {d}\vartheta _{\mathsf {P}}^{-,\varepsilon }}{\mathrm {d}\Theta ^{-,n,\varepsilon }_{\mathsf {P}}} \right)\sqrt{U(\nu -\genfrac{}{}{}2{1}{n}\delta _x)U(\nu )} \, \Pi _n(\mathrm {d}\nu ) \, \kappa ^{-,\varepsilon }_{\nu }(\mathrm {d}x) \\&=\mathcal {R}_{n,\varepsilon }\left(\mathsf {P},\vartheta _{\mathsf {P}}^{+,\varepsilon },\vartheta _{\mathsf {P}}^{-,\varepsilon }\right).\) 

Moreover, it is straightforward to check that
\( \int _{\Gamma \times \Gamma } \left(\sqrt{U_t(\eta )}-\sqrt{U_t(\nu )}\right)^2 \Pi _n(\mathrm {d}\nu ) \, \bar{\kappa }_{n}^{\varepsilon }(\nu ,\mathrm {d}\eta ) = \mathcal {D}_{n,\varepsilon }(\mathsf {P}), \) 
and therefore with \(J^{\pm }:=\vartheta _{\mathsf {P}}^{\pm ,\varepsilon }\)  we conclude
\( \mathcal {I}_{n,\varepsilon }(\mathsf {P},\mathsf {J}^+,\mathsf {J}^-)=0. \) 
Finally, note that by Lemma REF  and Remark  REF
\(\begin{aligned}\mathcal {E}\mathrm {nt}\left(\mathsf {J}^{\pm }|\Theta _{\mathsf {P}}^{n,+,\varepsilon }\right)&=\int _{\Gamma \times \mathcal {T}} \Upsilon \left(\frac{\mathrm {d}\vartheta _{\mathsf {P}}^{\pm ,\varepsilon }}{\mathrm {d}\Sigma },\frac{\mathrm {d}\vartheta _{\mathsf {P}}^{\pm ,\varepsilon }}{\mathrm {d}\Sigma },\frac{\mathrm {d}(\mathsf {T}^{n,\mp }_{\#}\vartheta _{\mathsf {P}}^{\mp ,\varepsilon })}{\mathrm {d}\Sigma }\right)\mathrm {d}\Sigma ,\\\mathcal {D}_{n,\varepsilon }&=2H^2(\vartheta _{\mathsf {P}}^{\pm ,\varepsilon },\mathsf {T}^{n,\mp }_{\#}\vartheta _{\mathsf {P}}^{\mp ,\varepsilon }),\end{aligned}\) 
for any dominating measure \(\Sigma \) , which are both non-negative, convex and vaguely lower-semicontinuous functionals of \(\vartheta _{\mathsf {P}}^{\pm ,\varepsilon },\mathsf {T}^{n,\mp }_{\#}\vartheta _{\mathsf {P}}^{\mp ,\varepsilon }\)  in \(\mathcal {M}_{loc}(\Gamma \times \mathcal {T})\) , see [3]}.
Next, we consider the sequence of pairs \((\mathsf {P}^{\varepsilon },\mathsf {J}^{\pm ,\varepsilon })\)  stemming from the regularized problems above, satisfying
\( \mathcal {I}_{n,\varepsilon }(\mathsf {P}^{\varepsilon },\mathsf {J}^{+,\varepsilon },\mathsf {J}^{-,\varepsilon })=0 \qquad \mbox{ for all }\varepsilon >0. \) 
As for a priori estimates, we have
\(\sup _{\varepsilon ,t\in [0,T]} \mathcal {E}\mathrm {nt}(\mathsf {P}_t^{\varepsilon }|\Pi _n) \le \mathcal {E}\mathrm {nt}(\bar{\mathsf {P}}|\Pi _n),\) 
and
\(\kappa ^{\pm ,\varepsilon }_{\nu }(\mathcal {T})\le \kappa ^{\pm }_{\nu }(\mathcal {T}) \qquad \mbox{ for all }\varepsilon >0.\) 
From the latter, it can be shown similarly as in Lemma REF  that we have the equicontinuity result
\(d_{TV,w}(\mathsf {P}^{\varepsilon }_t,\mathsf {P}^{\varepsilon }_s)\le 2 n \max \lbrace 1+\genfrac{}{}{}2{2}{n^2},2\rbrace  |t-s|.\) 
Here \(d_{TV,w}\)  is the weighted total variation-metric defined in (REF ) as
\(\begin{aligned}d_{TV,w}(\mathsf {P}^{\varepsilon }_t,\mathsf {P}^{\varepsilon }_s)&:=\int _{\Gamma } (1+\nu (\mathcal {T})^2)^{-1}\, \mathrm {d}|\mathsf {P}^1-\mathsf {P}^2|, \mbox{ for all }\varepsilon >0,\, \mbox{ for all }s,t\in [0,T].\end{aligned}\) 
Recall that \(d\)  is lower semicontinuous with respect to the narrow topology and convergence in \(d\)  implies narrow convergence on narrowly pre-compact sets. Since \(\mathcal {E}\mathrm {nt}(\mathsf {P}^{\varepsilon }_t|\Pi _n)\)  is bounded uniformly in \(\varepsilon \)  and \(t\)  and \(\mathcal {E}\mathrm {nt}(\cdot |\Pi _n)\)  is narrowly coercive we obtain by a standard Arzelá-Ascoli argument, up to choosing a subsequence, the existence of a curve \(t\mapsto \mathsf {P}_t\)  such that
\(\mathsf {P}_t^{\varepsilon } \rightarrow \mathsf {P}_t \quad \mbox{narrowly\; for all $t\in [0,T]$}.\) 
Note that by the estimate (REF ) and lower-semicontinuity of the entropy, we have that for every \(t\in [0,T]\) , the sequence of measures \(\mathsf {P}_t^{\varepsilon }\)  converge setwise to \(\mathsf {P}_t\)  and \(\mathcal {E}\mathrm {nt}(\mathsf {P}_t|\Pi _n)\le \mathcal {E}\mathrm {nt}(\bar{\mathsf {P}}|\Pi _n)<\infty \) . Moreover, \(\kappa ^{\pm ,\varepsilon }_{\nu } \nearrow \kappa ^{\pm }_{\nu }\)  as \(\varepsilon \rightarrow 0\)  for every \(\nu \) , and hence setwise convergence of \(\mathsf {P}^{\varepsilon }_t\)  implies setwise convergence on pre-compact sets of \(\Gamma \times \mathcal {T}\)  for
\( \vartheta _{\mathsf {P}^{\varepsilon }_t}^{\pm ,\varepsilon }(\mathrm {d}\nu ,\mathrm {d}x)=\mathsf {P}_t^{\varepsilon }(\mathrm {d}\nu )\kappa ^{\pm ,\varepsilon }[\nu ](\mathrm {d}x), \) 
see e.g. [1]} for the case of set-wise convergence for bounded jump kernels. In particular we have the vague convergence
\(\vartheta ^{\pm ,\varepsilon }_{\mathsf {P}_t^{\varepsilon }} \rightarrow \vartheta _{\mathsf {P}_t}^\pm ,\qquad \mathsf {T}^{n,\pm }_{\#}\vartheta ^{\pm ,\varepsilon }_{\mathsf {P}_t^{\varepsilon }} \rightarrow \mathsf {T}^{n,\pm }_{\#}\vartheta _{\mathsf {P}_t}^\pm .\) 

It is straightforward to check that we can pass to the limit in the continuity equation (REF ), and in particular, derive that \(\mathsf {P}\)  is a weak solution to the unregularized problem.
Finally, recall that \(\mathcal {F}_n(\mathsf {P}_T)\)  is convex in and narrowly lower semicontinuous in \(\mathsf {P}^{\varepsilon }_T\) , and as shown above the action \(\mathcal {R}^{\varepsilon }_n\)  is jointly convex and lower semicontinuous in \((\vartheta ^{\pm ,\varepsilon }_{\mathsf {P}^{\varepsilon }},\mathsf {T}^{n,\mp }_{\#}\vartheta ^{\mp ,\varepsilon }_{\mathsf {P}^{\varepsilon }})\) . Proceeding as in Remark REF , we also find that the Fisher information is jointly convex and lower semicontinuous in \((\vartheta ^{\pm ,\varepsilon }_{\mathsf {P}^{\varepsilon }},\mathsf {T}^{n,\mp }_{\#}\vartheta ^{\mp ,\varepsilon }_{\mathsf {P}^{\varepsilon }})\)  if \(\mathsf {P}^{\varepsilon }\)  are contained in sub-level sets of \(\mathcal {F}_n\) .
Therefore, we conclude that
\(\mathcal {I}_{n}(\mathsf {P})&\le \liminf _{\varepsilon \rightarrow 0} \left( \int _0^T \mathcal {R}_{n,\varepsilon }(\mathsf {P}^{\varepsilon }_t,\mathsf {J}_t^{+,\varepsilon },\mathsf {J}_t^{-,\varepsilon }) \, \mathrm {d}t+\mathcal {F}_n(\mathsf {P}_T^{\varepsilon })-\mathcal {F}_n(\bar{\mathsf {P}})+\int _0^T \mathcal {D}_{n,\varepsilon }(\mathsf {P}^ {\varepsilon }_t) \, \mathrm {d}t \right)\\ &=\mathcal {I}_{n,\varepsilon }(\mathsf {P}^{\varepsilon },\mathsf {J}^{+,\varepsilon },\mathsf {J}^{-,\varepsilon }) =0,\) 

thus establishing the claim.
